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2 2 2 2 1 0 0 1 } 1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 233 240 "In this Maple file, we \+ compute the evolution equations for the Painlev\351 2 equations using \+ the compatibility equation of the Lax system. We also obtain the expre ssion of the Lax matrices in the geometric gauge without apparent sing ularities." }{TEXT 233 0 "" }}{PARA 0 "" 0 "" {TEXT 233 90 "The operat or is \\hbar (alpha111\\partial_\{t_\{1^\{(1)\},1\} +alpha121\\partial _\{t_\{1^\{(2)\},1\})) " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 234 56 "Lax \+ matrices in the oper gauge from previous Maple files" }{TEXT 234 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 212 102 "Summary of previous files: We have the expression for some coefficients of the Lax matrix L and of \+ A." }{TEXT 212 0 "" }}}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 9 "rest art:\n" }{MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 19 "P0 11 := t010+t020;\n" }{MPLTEXT 1 0 19 "P111 := t110+t120;\n" }{MPLTEXT 1 0 19 "P121 := t111+t121;\n" }{MPLTEXT 1 0 19 "P142 := t111*t121;\n" }{MPLTEXT 1 0 29 "P132 := t110*t121+t111*t120;\n" }{MPLTEXT 1 0 19 "P0 22 := t010*t020;\n" }{MPLTEXT 1 0 59 "CoherenceEquation1:=tinfty10+tin fty20+t010+t020+t110+t120; " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 49 "P1 := unapply(P011/x+P121/(x-1)^2+P111/(x-1),x);\n" }{MPLTEXT 1 0 83 "P2 := unapply(P022/x^2+P012/x+P142/(x-1)^4+P132/(x-1)^3+P122/(x-1)^2+P11 2/(x-1),x);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 80 "tdP2:=unapply(P2(l ambda)-P012/lambda-P112/(lambda-1)-P122/(lambda-1)^2,lambda);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 "c1:=(alpha111*t121-alpha121*t111 )/(t111-t121);" }{TEXT 212 1 "\n" }{MPLTEXT 1 0 47 "mu:=(alpha111-alph a121)*q*(q-1)^2/(t111-t121);\n" }{MPLTEXT 1 0 45 "nu0:=(alpha111-alpha 121)*(q-1)^2/(t111-t121);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 49 "nuMi nus1:=(alpha111-alpha121)*(q-1)/(t111-t121);\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 10 "C12:=-a2:\n" }{MPLTEXT 1 0 10 "C11:=-a1:\n" }{MPLTEXT 1 0 9 "C01:=-a0:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 48 "c11:=(alpha1 11*t121-alpha121*t111)/(t111-t121);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "dP1dlambda:=unapply(diff(P1(lambda),lambda),lambda):\n" } {MPLTEXT 1 0 53 "dP2dlambda:=unapply(diff(P2(lambda),lambda),lambda): \n" }{MPLTEXT 1 0 18 "L:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 11 "L[1,1]:=0 :\n" }{MPLTEXT 1 0 11 "L[1,2]:=1:\n" }{MPLTEXT 1 0 8 "L[2,1]:=" } {MPLTEXT 1 0 91 "-P2(lambda)+(P012-a0)/lambda+(P122-a2)/(lambda-1)^2+ \+ (P112-a1)/(lambda-1) -h*p/(lambda-q):\n" }{MPLTEXT 1 0 9 "L[2,2]:= " } {MPLTEXT 1 0 11 "P1(lambda)-" }{MPLTEXT 1 0 23 "h/lambda-2*h/(lambda-1 )" }{MPLTEXT 1 0 15 "+h/(lambda-q):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 "Eq1:=" }{MPLTEXT 1 228 11 "a0+a1+h*p;\n" }{MPLTEXT 1 229 6 "Eq2 := " }{MPLTEXT 1 230 44 "h*p*q+t010*t020+a1+a2-(h+tinfty20)*tinfty10;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 7 "c0:=0:\n" }{MPLTEXT 1 0 18 "A:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 9 "A[1,1]:= " } {MPLTEXT 1 0 33 "c1/(lambda-1) +c0+rho/(lambda-q):" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 42 "A[1,2]:= nuMinus1*lambda+nu0+mu/(lambda-q)" } {MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "A[2,1]:= AA2 1(lambda):\n" }{MPLTEXT 1 0 23 "A[2,2]:= AA22(lambda):\n" }{MPLTEXT 1 0 26 "dAdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 88 "for i from 1 to 2 do for j from 1 to 2 do dAdlambda[i,j]:=diff(A[i,j],lambda): od: od: \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 3 "L;\n" }{MPLTEXT 1 0 3 "A;\n" }{MPLTEXT 1 0 15 "P012 := -P112;\n" }{MPLTEXT 1 0 43 "P122 := -t010*t 020+tinfty10*tinfty20-P112;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 37 " Q2:=unapply(-p*(q-0)*(q-1)^2,lambda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "J:=Matrix(2,2,0):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "J[1, 1]:=1:\n" }{MPLTEXT 1 0 11 "J[1,2]:=0:\n" }{MPLTEXT 1 0 31 "J[2,1]:=Q2 (lambda)/(lambda-q):\n" }{MPLTEXT 1 0 45 "J[2,2]:=(lambda-0)^1*(lambda -1)^2/(lambda-q):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 "dJdlambda:=M atrix(2,2,0):\n" }{MPLTEXT 1 0 87 "for i from 1 to 2 do for j from 1 t o 2 do dJdlambda[i,j]:=diff(J[i,j],lambda): od: od:\n" }{MPLTEXT 1 0 3 "J:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "LJ:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 12 "LJ[1,1]:=0:\n" }{MPLTEXT 1 0 12 "LJ[1,2]:=0:\n" } {MPLTEXT 1 0 104 "LJ[2,2]:=diff(J[2,2],q)*Lq+diff(J[2,2],p)*Lp+h*alpha 111*diff(J[2,2],t111)+h*alpha121*diff(J[2,2],t121):\n" }{MPLTEXT 1 0 104 "LJ[2,1]:=diff(J[2,1],q)*Lq+diff(J[2,1],p)*Lp+h*alpha111*diff(J[2, 1],t111)+h*alpha121*diff(J[2,1],t121):\n" }{MPLTEXT 1 0 3 "LJ:" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 79 "checkL:=sim plify(Multiply(Multiply(J,L),J^(-1))+h*Multiply(dJdlambda,J^(-1))):\n" }{MPLTEXT 1 0 70 "checkA:=simplify(Multiply(Multiply(J,A),J^(-1))+Mul tiply(LJ,J^(-1))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I%P011G6\",&I%t010GF$\"\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P111G6\",&I%t110GF$\"\"\"I%t120GF$ F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P121G6\",&I%t111GF$\"\"\"I%t121 GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P142G6\"*&I%t111GF$\"\"\"I%t 121GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P132G6\",&*&I%t110GF$\"\" \"I%t121GF$F(F(*&I%t111GF$F(I%t120GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P022G6\"*&I%t010GF$\"\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation1G6\",.I)tinfty10GF$\"\"\"I)tinfty 20GF$F'I%t010GF$F'I%t020GF$F'I%t110GF$F'I%t120GF$F'" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$,(*&,& I%t010GF$\"\"\"I%t020GF$F/F/9$!\"\"F/*&,&I%t111GF$F/I%t121GF$F/F/,&F1F /F2F/!\"#F/*&,&I%t110GF$F/I%t120GF$F/F/F7F2F/F$F$F$" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I#P2G6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$,.*(I% t010GF$\"\"\"I%t020GF$F.9$!\"#F.*&I%P012GF$F.F0!\"\"F.*(I%t111GF$F.I%t 121GF$F.,&F0F.F4F.!\"%F.*&,&*&I%t110GF$F.F7F.F.*&F6F.I%t120GF$F.F.F.F8 !\"$F.*&I%P122GF$F.F8F1F.*&I%P112GF$F.F8F4F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$, (*(I%t010GF$\"\"\"I%t020GF$F.9$!\"#F.*(I%t111GF$F.I%t121GF$F.,&F0F.!\" \"F.!\"%F.*&,&*&I%t110GF$F.F4F.F.*&F3F.I%t120GF$F.F.F.F5!\"$F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\"*&,&*&I%t111GF$\"\"\"I)alpha1 21GF$F)!\"\"*&I%t121GF$F)I)alpha111GF$F)F)F),&F(F)F-F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I#muG6\"**,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\" F(I\"qGF$F(,&F+F(F*F(\"\"#,&I%t111GF$F(I%t121GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu0G6\"*(,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F(, &I\"qGF$F(F*F(\"\"#,&I%t111GF$F(I%t121GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuMinus1G6\"*(,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F (,&I\"qGF$F(F*F(F(,&I%t111GF$F(I%t121GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$c11G6\"*&,&*&I%t111GF$\"\"\"I)alpha121GF$F)!\"\"*&I%t1 21GF$F)I)alpha111GF$F)F)F),&F(F)F-F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$Eq1G6\",(*&I\"hGF$\"\"\"I\"pGF$F(F(I#a0GF$F(I#a1GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$Eq2G6\",,*(I\"hGF$\"\"\"I\"pGF$F(I\"qGF$F(F(* &I%t010GF$F(I%t020GF$F(F(I#a1GF$F(I#a2GF$F(*&,&F'F(I)tinfty20GF$F(F(I) tinfty10GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+mo dulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$m trGF$6'-I$mtdGF$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvari antGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rows panGQ\"1F'/%+columnspanGFJ-F56(-F86%FJF;F>FAFDFFFHFKFAFDFF-F26'-F56(-F ,66-I#moGF$6.Q*&uminus0;F'F;F>/%&fenceGQ&falseF'/%*separatorGFgn/%)str etchyGFgn/%*symmetricGFgn/%(largeopGFgn/%.movablelimitsGFgn/%'accentGF gn/%'lspaceGQ,0.2222222emF'/%'rspaceGFfo-I&mfracGF$6)-F,6%-I#miGF$6&Q% t010F'/%'italicGQ%trueF'F;/F?Q'italicF'-FX6-Q1⁢F'F>FenF hnFjnF\\oF^oF`oFbo/FeoQ&0.0emF'/FhoF[q-F_p6&Q%t020F'FbpF;Fep-F,6#-I%ms upGF$6%-F_p6&Q)λF'/FcpFgnF;F>-F86%Q\"2F'F;F>/%1superscriptshift GF:/%.linethicknessGFJ/%+denomalignGQ'centerF'/%)numalignGFbr/%)bevell edGFgnF;-FX6.Q(−F'F;F>FenFhnFjnF\\oF^oF`oFboFdoFgo-Fjo6)-F,6#-F_ p6&Q%P012F'FbpF;Fep-F,6#FeqF^rF`rFcrFerF;Fgr-Fjo6)-F,6%-F_p6&Q%t111F'F bpF;FepFgp-F_p6&Q%t121F'FbpF;Fep-F,6#-Fcq6%-F#6%-F,6%FeqFgrFOF;F>-F86% Q\"4F'F;F>F\\rF^rF`rFcrFerF;Fgr-Fjo6)-F,6#-F,6%-F,6%-F_p6&Q%t110F'FbpF ;FepFgpFjs-FX6.Q\"+F'F;F>FenFhnFjnF\\oF^oF`oFboFdoFgo-F,6%FgsFgp-F_p6& Q%t120F'FbpF;Fep-F,6#-Fcq6%Fat-F86%Q\"3F'F;F>F\\rF^rF`rFcrFerF;Fgr-Fjo 6)-F,6#-F_p6&Q%P122F'FbpF;Fep-F,6#-Fcq6%FatFiqF\\rF^rF`rFcrFerF;Fgr-Fj o6)-F,6#-F_p6&Q%P112F'FbpF;Fep-F,6#FctF^rF`rFcrFerF;Fcu-Fjo6)-F,6#-F,6 %F^sFgr-F_p6&Q#a0F'FbpF;FepFasF^rF`rFcrFerF;Fcu-Fjo6)-F,6#-F,6%FfvFgr- F_p6&Q#a2F'FbpF;FepFivF^rF`rFcrFerF;Fcu-Fjo6)-F,6#-F,6%FawFgr-F_p6&Q#a 1F'FbpF;FepFdwF^rF`rFcrFerF;Fgr-Fjo6)-F,6%-F_p6&Q\"hF'FbpF;FepFgp-F_p6 &Q\"pF'FbpF;Fep-F,6#-F,6%FeqFgr-F_p6&Q\"qF'FbpF;FepF^rF`rFcrFerF;FAFDF FFHFK-F56(-F,6--Fjo6)-F,6#-F,6%F^pFcuF]qFasF^rF`rFcrFerF;Fcu-Fjo6)-F,6 #-F,6%FgsFcuFjsFivF^rF`rFcrFerF;Fcu-Fjo6)-F,6#-F,6%F`uFcuFhuFdwF^rF`rF crFerF;Fgr-Fjo6)-F,6#FeyFasF^rF`rFcrFerF;Fgr-Fjo6)-F,6%FiqFgpFeyFdwF^r F`rFcrFerF;Fcu-Fjo6)Fj[lF[zF^rF`rFcrFerF;FAFDFFFHFKFAFDFF/%&alignGQ%ax isF'/FBQ)baselineF'/FEFbr/FGQ'|frleft|hrF'/%/alignmentscopeGFdp/%,colu mnwidthGQ%autoF'/%&widthGF^]l/%+rowspacingGQ&1.0exF'/%.columnspacingGQ &0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFi]l/%&frameGFi]l/%-framesp acingGQ,0.4em~0.5exF'/%*equalrowsGFgn/%-equalcolumnsGFgn/%-displaystyl eGFgn/%%sideGQ&rightF'/%0minlabelspacingGFf]lF;F>/%%openGQ\"[F'/%&clos eGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$ mtdGF$6(-F,6%-I&mfracGF$6)-F,6#-F,6&-I#moGF$6.Q*&uminus0;F'/%+foregrou ndGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorG FL/%)stretchyGFL/%*symmetricGFL/%(largeopGFL/%.movablelimitsGFL/%'acce ntGFL/%'lspaceGQ,0.2222222emF'/%'rspaceGFen-F,6%-I#miGF$6&Q%t111F'/%'i talicGQ%trueF'FD/FHQ'italicF'-FA6-Q1⁢F'FGFJFMFOFQFSFUFW /FZQ&0.0emF'/FgnFgo-F[o6&Q+α121F'F^oFDFao-FA6.Q\"+F'FDFGFJFMFOFQ FSFUFWFYFfn-F,6%-F[o6&Q%t121F'F^oFDFaoFco-F[o6&Q+α111F'F^oFDFao- F,6%-F#6%-F,6%Fjn-FA6.Q(−F'FDFGFJFMFOFQFSFUFWFYFfnFapFDFGFco-F#6 %-F,6%-F[o6&Q)λF'/F_oFLFDFGF]q-I#mnGF$6%Q\"1F'FDFGFDFG/%.lineth icknessGF[r/%+denomalignGQ'centerF'/%)numalignGF`r/%)bevelledGFLFDF\\p -F:6)-F,6#-F[o6&Q&ρF'FgqFDFG-F,6#-F,6%FdqF]q-F[o6&Q\"qF'F^oFDFaoF \\rF^rFarFcrFD/%)rowalignGQ!F'/%,columnalignGFes/%+groupalignGFes/%(ro wspanGF[r/%+columnspanGF[r-F56(-F,6'-F:6)-F,6'-F#6%-F,6%FdpF]qFioFDFGF co-F#6%-F,6%F`sF]qFhqFDFGFcoFdq-F,6#F[qF\\rF^rFarFcrFDF\\p-F:6)-F,6%Ff tFco-I%msupGF$6%Fjt-Fiq6%Q\"2F'FDFG/%1superscriptshiftGQ\"0F'F^uF\\rF^ rFarFcrFDF\\p-F:6)-F,6'FftFcoF`sFcoFdu-F,6%FipFco-F#6%F^sFDFGF\\rF^rFa rFcrFDFcsFfsFhsFjsF\\tFcsFfsFhs-F26'-F56(-F,6%-F[o6&Q%AA21F'F^oFDFao-F A6-Q0⁡F'FGFJFMFOFQFSFUFWFfoFho-F#6%-F,6#FdqFDFGFcsFfsFhs FjsF\\t-F56(-F,6%-F[o6&Q%AA22F'F^oFDFaoF^wFawFcsFfsFhsFjsF\\tFcsFfsFhs /%&alignGQ%axisF'/FdsQ)baselineF'/FgsF`r/FisQ'|frleft|hrF'/%/alignment scopeGF`o/%,columnwidthGQ%autoF'/%&widthGFhx/%+rowspacingGQ&1.0exF'/%. columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFcy/%&frameG Fcy/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFL/%-equalcolumnsGFL/%- displaystyleGFL/%%sideGQ&rightF'/%0minlabelspacingGF`yFDFG/%%openGQ\"[ F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P012G6\",$I%P112 GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P122G6\",(*&I%t010GF$\"\" \"I%t020GF$F(!\"\"*&I)tinfty10GF$F(I)tinfty20GF$F(F(I%P112GF$F*" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 234 73 "Solving the compatibility equati ons to obtain the Hamiltonian evolutions." }{TEXT 234 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 233 69 "The compatibility equation is \\mathcal \{L\}L=h\\partial_\\lambda A+[A,L]\n" }{TEXT 233 111 "Since the first \+ line of L is trivial, we may easily obtain A[2,1] et A[2,2] to obtain \+ the full expression for A" }{TEXT 233 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "LL:=h*dAdlambda+(Multiply(A,L)-Multiply(L,A)):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "Entry11:=LL[1,1]:\n" }{MPLTEXT 1 0 18 "Entry12:=LL[1,2]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "AA 21:=unapply(solve(Entry11=0,AA21(lambda)),lambda):\n" }{MPLTEXT 1 0 41 "AA21bis:=h*dAdlambda[1,1]+A[1,2]*L[2,1]:\n" }{MPLTEXT 1 0 32 "simp lify(AA21(lambda)-AA21bis);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 53 "AA22:=unapply(solve(Entry12=0,AA22(lambda)),lamb da):\n" }{MPLTEXT 1 0 48 "AA22bis:=h*dAdlambda[1,2]+A[1,1]+A[1,2]*L[2, 2]:\n" }{MPLTEXT 1 0 32 "simplify(AA22(lambda)-AA22bis);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "simplify(Entry11);\n" }{MPLTEXT 1 0 19 "s implify(Entry12);\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 46 "L:=h*dAdlam bda+(Multiply(A,L)-Multiply(L,A)):\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 233 95 "We now compute the acti on of \\mathcal\{L\} on L[2,2] et L[2,1] to obtain the evolution equa tions" }{TEXT 233 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 26 "Evoluti on of entry L_\{2,2\}" }{TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry22:=simplify(LL[2,2]);\n" }{MPLTEXT 1 0 77 "Entr y22TermLambdaMinusqCube:=factor(residue(Entry22*(lambda-q)^2,lambda=q) );\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaMinusqSquare:=factor(residue( Entry22*(lambda-q),lambda=q));\n" }{MPLTEXT 1 0 60 "Entry22TermLambdaM inusq:=factor(residue(Entry22,lambda=q));\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 77 "Entry22TermLambdaInfty1:=factor(-residue(Entry22/lamb da^2,lambda=infinity));\n" }{MPLTEXT 1 0 74 "Entry22TermLambdaInfty0:= factor(-residue(Entry22/lambda,lambda=infinity));" }{MPLTEXT 1 0 1 "\n " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "Entry22TermLambda" }{MPLTEXT 1 0 36 "1OrderMinus1:=factor(residue(Entry22" }{MPLTEXT 1 0 13 "/(lamb da-1)^0" }{MPLTEXT 1 0 13 ",lambda=1));\n" }{MPLTEXT 1 0 17 "Entry22Te rmLambda" }{MPLTEXT 1 0 37 "1OrderMinus2:=factor(residue(Entry22*" } {MPLTEXT 1 0 10 "(lambda-1)" }{MPLTEXT 1 0 12 ",lambda=1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 78 "Entry22TermLambda1OrderMinus3:=factor(re sidue(Entry22*(lambda-1)^2,lambda=1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Entry22TermLambda0OrderMinus1:=factor(re sidue(Entry22/(lambda)^0,lambda=0));\n" }{MPLTEXT 1 0 75 "Entry22TermL ambda0OrderMinus2:=factor(residue(Entry22*(lambda),lambda=0));\n" } {MPLTEXT 1 0 76 "Entry22TermLambda0OrderMinus3:=factor(residue(Entry22 *(lambda)^2,lambda=0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 97 "simplify( Entry22-(Entry22TermLa mbdaMinusqSquare/(lambda-q)^2+Entry22TermLambdaMinusq/(lambda-q)\n" } {MPLTEXT 1 0 25 "+Entry22TermLambdaInfty0+" }{MPLTEXT 1 0 42 "Entry22T ermLambda1OrderMinus2/(lambda-1)^2" }{MPLTEXT 1 0 5 ") );\n" }{MPLTEXT 1 0 8 "L[2,2];\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Entry22G6\"*,I\"hGF$\"\"\",0*&,**(,&I)alpha 111GF$F'I)alpha121GF$!\"\"F',,F&F'I%t010GF$F/I%t020GF$F/I%t110GF$F/I%t 120GF$F/F'I'lambdaGF$\"\"#F'*(F,F'F0F'F5F'!\"#*&,0F&F'F1F/F2F/F3F/I%t1 11GF$F'F4F/I%t121GF$F/F'F-F'F'*&F.F',0F&F'F1F/F2F/F3F/F;F/F4F/FII>Entry22TermLambdaMinusqSquareG6\"*(,hn*(I\"hGF$\"\" \"I\"qGF$\"\"#I)alpha111GF$F)F)*(F(F)F*F+I)alpha121GF$F)!\"\"*(F*F+I%t 010GF$F)F,F)F/*(F*F+F1F)F.F)F)*(F*F+I%t020GF$F)F,F)F/*(F*F+F4F)F.F)F)* (F*F+I%t110GF$F)F,F)F/*(F*F+F7F)F.F)F)*(F*F+I%t120GF$F)F,F)F/*(F*F+F:F )F.F)F)*(F(F)F*F)F,F)F/*(F(F)F*F)F.F)F)*(F*F)F1F)F,F)F+*(F*F)F1F)F.F)! \"#*(F*F)F4F)F,F)F+*(F*F)F4F)F.F)F@*(F*F)F7F)F,F)F)*(F*F)F7F)F.F)F/*(F *F)I%t111GF$F)F,F)F/*(F*F)FFF)F.F)F)*(F*F)F:F)F,F)F)*(F*F)F:F)F.F)F/*( F*F)I%t121GF$F)F,F)F/*(F*F)FKF)F.F)F)*&I$rhoGF$F)FFF)F@*&FNF)FKF)F+*&F 1F)F,F)F/*&F1F)F.F)F)*&F4F)F,F)F/*&F4F)F.F)F)F)F(F),&FFF)FKF/F/" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaMinusqG6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty1G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty0G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I>Entry22TermLambda1OrderMinus1G6\"\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I>Entry22TermLambda1OrderMinus2G6 \"*&,&I)alpha111GF$\"\"\"I)alpha121GF$F(F(I\"hGF$F(" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I>Entry22TermLambda1OrderMinus3G6\"\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I>Entry22TermLambda0OrderMinus1G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I>Entry22TermLambda0OrderMinus2G6\"\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I>Entry22TermLambda0OrderMinus3G6 \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&,&I%t010G6\"\"\"\"I%t020GF&F'F'I'lambdaGF&!\"\"F'*&,& I%t111GF&F'I%t121GF&F'F',&F)F'F*F'!\"#F'*&,&I%t110GF&F'I%t120GF&F'F'F/ F*F'*&I\"hGF&F'F)F*F**&F6F'F/F*F0*&F6F',&F)F'I\"qGF&F*F*F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 233 167 "Since the deformation operator is \\hba r (alpha111\\partial_\{t_\{1^\{(1)\},1\} +alpha121\\partial_\{t_\{1^\{ (2)\},1\}) it acts on the double pole and gives h*(alpha111+alpha121). \n" }{TEXT 233 118 "On the simple pole at lambda=q we find h*L[q] that should correspond to the double pole of the compatibility equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Equation1:=Entry22TermLam bda1OrderMinus2- h*(alpha111+alpha121);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation1G6\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Lq:=factor(Entry22TermLambdaMinusqSquare/h):\n" }{MPLTEXT 1 0 111 "Lqbis:=-2*rho+mu/q/(q-1)^2*( (h-t120-t010-t020-t110)*q^2-q*(h-2*t010- 2*t020-t110+t111-t120+t121)-(t010+t020));\n" }{MPLTEXT 1 0 28 "factor( simplify(Lq-Lqbis));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&LqbisG6\",& I$rhoGF$!\"#*(,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F+,&I%t111GF$F+I% t121GF$F-F-,**&,,I\"hGF$F+I%t010GF$F-I%t020GF$F-I%t110GF$F-I%t120GF$F- F+I\"qGF$\"\"#F+*&F9F+,0F4F+F5F'F6F'F7F-F/F+F8F-F0F+F+F-F5F-F6F-F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 233 32 "Evolution of \\mathcal\{L\}[L[2,1]]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry21:=simplify(LL[2,1]):\n" }{MPLTEXT 1 0 77 " Entry21TermLambdaMinusqCube:=factor(residue(Entry21*(lambda-q)^2,lambd a=q));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaMinusqSquare:=factor(resi due(Entry21*(lambda-q),lambda=q));\n" }{MPLTEXT 1 0 60 "Entry21TermLam bdaMinusq:=factor(residue(Entry21,lambda=q));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty1:=factor(-residue(Entry21/lam bda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "Entry21TermLambdaInfty0: =factor(-residue(Entry21/lambda,lambda=infinity));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 68 "Entry21TermLambdaZero2:=factor(residue(Entry21* (lambda),lambda=0));\n" }{MPLTEXT 1 0 58 "Entry21TermLambdaZero1:=fact or(residue(Entry21,lambda=0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 69 "Entry21TermLambdaUn5:=factor(residue(Entry21*(la mbda-1)^4,lambda=1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 "Entry21T ermLambdaUn4:=factor(residue(Entry21*(lambda-1)^3,lambda=1));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 "Entry21TermLambdaUn3:=factor(res idue(Entry21*(lambda-1)^2,lambda=1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 67 "Entry21TermLambdaUn2:=factor(residue(Entry21*(lambda-1),lambda =1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 56 "Entry21TermLambdaUn1:=fa ctor(residue(Entry21,lambda=1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 138 "simplify( Entry21-(Entry21TermLambdaMinusqCu be/(lambda-q)^3+Entry21TermLambdaMinusqSquare/(lambda-q)^2+Entry21Term LambdaMinusq/(lambda-q)\n" }{MPLTEXT 1 0 56 "+Entry21TermLambdaInfty0+ Entry21TermLambdaInfty1*lambda\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 30 "Entry21TermLambdaZero1/lambda+" }{MPLTEXT 1 0 31 "Entry21TermLambd aUn1/(lambda-1)" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 ") );\n" } {MPLTEXT 1 0 8 "L[2,1];\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">II>Entry21TermLambdaMinusqSquareG6\"*,I\"hGF$\"\" \",`u**F&F'I\"pGF$F'I\"qGF$\"\"&I)alpha111GF$F'\"\"#**F&F'F*F'F+F,I)al pha121GF$F'!\"#*(I#a0GF$F'F+F,F-F'F.*(F3F'F+F,F0F'F1*(I#a1GF$F'F+F,F-F 'F.*(F6F'F+F,F0F'F1**F&F'F*F'F+\"\"%F-F'!\"(**F&F'F*F'F+F9F0F'\"\"(**F +F9I%t010GF$F'I%t020GF$F'F-F'F.**F+F9F>F'F?F'F0F'F1*(F3F'F+F9F-F'!\")* (F3F'F+F9F0F'\"\")*(F6F'F+F9F-F'!\"'*(F6F'F+F9F0F'\"\"'*(I#a2GF$F'F+F9 F-F'F.*(FJF'F+F9F0F'F1**F&F'F*F'F+\"\"$F-F'\"\"***F&F'F*F'F+FMF0F'!\"* **F+FMF>F'F?F'F-F'FB**F+FMF>F'F?F'F0F'FD**F+FMI%t110GF$F'I%t121GF$F'F- F'F.**F+FMFTF'FUF'F0F'F1**F+FMI%t111GF$F'I%t120GF$F'F-F'F.**F+FMFXF'FY F'F0F'F1*(F3F'F+FMF-F'\"#7*(F3F'F+FMF0F'!#7*(F6F'F+FMF-F'FH*(F6F'F+FMF 0F'FF*(FJF'F+FMF-F'!\"%*(FJF'F+FMF0F'F9**F&F'F*F'F+F.F-F'!\"&**F&F'F*F 'F+F.F0F'F,**F&F'F+F.I$rhoGF$F'FXF'!\"$**F&F'F+F.FboF'FUF'FM**F+F.FboF 'F>F'FXF'F'**F+F.FboF'F>F'FUF'!\"\"**F+F.FboF'F?F'FXF'F'**F+F.FboF'F?F 'FUF'Fgo**F+F.FboF'FTF'FXF'F'**F+F.FboF'FTF'FUF'Fgo**F+F.FboF'FXF'FYF' F'**F+F.FboF'FYF'FUF'Fgo**F+F.F>F'F?F'F-F'Ffn**F+F.F>F'F?F'F0F'Fhn**F+ F.FTF'FUF'F-F'F1**F+F.FTF'FUF'F0F'F.**F+F.FXF'FYF'F-F'F1**F+F.FXF'FYF' F0F'F.**F+F.FXF'FUF'F-F'F.**F+F.FXF'FUF'F0F'F1*(F3F'F+F.F-F'FB*(F3F'F+ F.F0F'FD*(F6F'F+F.F-F'F1*(F6F'F+F.F0F'F.*(FJF'F+F.F-F'F.*(FJF'F+F.F0F' F1**F&F'F*F'F+F'F-F'F'**F&F'F*F'F+F'F0F'Fgo**F&F'F+F'FboF'FXF'F9**F&F' F+F'FboF'FUF'F\\o**F+F'FboF'F>F'FXF'F1**F+F'FboF'F>F'FUF'F.**F+F'FboF' F?F'FXF'F1**F+F'FboF'F?F'FUF'F.**F+F'FboF'FTF'FXF'Fgo**F+F'FboF'FTF'FU F'F'*(F+F'FboF'FXF.F'**F+F'FboF'FXF'FYF'Fgo**F+F'FboF'FYF'FUF'F'*(F+F' FboF'FUF.Fgo**F+F'F>F'F?F'F-F'FB**F+F'F>F'F?F'F0F'FD*(F3F'F+F'F-F'F.*( F3F'F+F'F0F'F1*(F&F'FboF'FXF'Fgo*(F&F'FboF'FUF'F'*(FboF'F>F'FXF'F'*(Fb oF'F>F'FUF'Fgo*(FboF'F?F'FXF'F'*(FboF'F?F'FUF'Fgo*(F>F'F?F'F-F'F.*(F>F 'F?F'F0F'F1F',&F+F'FgoF'F1,&FXF'FUFgoFgoF+Fgo" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaMinusqG6\",$*,,\\w**I\"hGF$\"\"\"I\"p GF$F*I\"qGF$\"\"'I)alpha111GF$F*F***F)F*F+F*F,F-I)alpha121GF$F*!\"\"*( I#a0GF$F*F,F-F.F*F**(F3F*F,F-F0F*F1*(I#a1GF$F*F,F-F.F*F**(F6F*F,F-F0F* F1**F)F*F+F*F,\"\"&F.F*!\"%**F)F*F+F*F,F9F0F*\"\"%**F,F9I%t010GF$F*I%t 020GF$F*F.F*\"\"#**F,F9F>F*F?F*F0F*!\"#*(F3F*F,F9F.F*!\"&*(F3F*F,F9F0F *F9*(F6F*F,F9F.F*!\"$*(F6F*F,F9F0F*\"\"$*(I#a2GF$F*F,F9F.F*F@*(FKF*F,F 9F0F*FB**F)F*F+F*F,FF*F?F*F.F*!#5 **F,FF*F?F*F0F*\"#5**F,FF*FYF*F***F,FIF_oF*F>F*FVF*F1**F,FIF_oF*F?F*FYF*F***F, FIF_oF*F?F*FVF*F1**F,FIF_oF*FUF*FYF*F***F,FIF_oF*FUF*FVF*F1**F,FIF_oF* FYF*FZF*F***F,FIF_oF*FZF*FVF*F1**F,FIF>F*F?F*F.F*\"#?**F,FIF>F*F?F*F0F *!#?**F,FIFUF*FVF*F.F*FG**F,FIFUF*FVF*F0F*FI**F,FIFYF*FZF*F.F*FG**F,FI FYF*FZF*F0F*FI**F,FIFYF*FVF*F.F*F<**F,FIFYF*FVF*F0F*F:*(F3F*F,FIF.F*FQ *(F3F*F,FIF0F*FS*(F6F*F,FIF.F*F1*(F6F*F,FIF0F*F**(FKF*F,FIF.F*F@*(FKF* F,FIF0F*FB**F)F*F+F*F,F@F.F*F***F)F*F+F*F,F@F0F*F1**F)F*F,F@F_oF*FYF*F 9**F)F*F,F@F_oF*FVF*FD**F)F*F,F@FYF*F0F*F1**F)F*F,F@FVF*F.F*F***F,F@F_ oF*F>F*FYF*FG**F,F@F_oF*F>F*FVF*FI**F,F@F_oF*F?F*FYF*FG**F,F@F_oF*F?F* FVF*FI**F,F@F_oF*FUF*FYF*F1**F,F@F_oF*FUF*FVF*F**(F,F@F_oF*FYF@F@**F,F @F_oF*FYF*FZF*F1**F,F@F_oF*FZF*FVF*F**(F,F@F_oF*FVF@FB**F,F@F>F*F?F*F. F*F^p**F,F@F>F*F?F*F0F*F\\p*(F3F*F,F@F.F*F9*(F3F*F,F@F0F*FD**F)F*F,F*F _oF*FYF*FG**F)F*F,F*F_oF*FVF*FI**F,F*F_oF*F>F*FYF*FI**F,F*F_oF*F>F*FVF *FG**F,F*F_oF*F?F*FYF*FI**F,F*F_oF*F?F*FVF*FG**F,F*F>F*F?F*F.F*FS**F,F *F>F*F?F*F0F*FQ*(F3F*F,F*F.F*F1*(F3F*F,F*F0F*F**(F)F*F_oF*FYF*F**(F)F* F_oF*FVF*F1*(F_oF*F>F*FYF*F1*(F_oF*F>F*FVF*F**(F_oF*F?F*FYF*F1*(F_oF*F ?F*FVF*F**(F>F*F?F*F.F*FB*(F>F*F?F*F0F*F@F*F)F*,&F,F*F1F*FG,&FYF*FVF1F 1F,FBF1" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty1G6 \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty0G6 \"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7Entry21TermLambdaZero2G6\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7Entry21TermLambdaZero1G6\",$ **I\"hGF$\"\"\"I\"qGF$!\"#,N**F'F(F)\"\"#I%t111GF$F(I)alpha121GF$F(!\" \"**F'F(F)F-I%t121GF$F(I)alpha111GF$F(F(**F)F-I%t010GF$F(I%t020GF$F(F3 F(F***F)F-F5F(F6F(F/F(F-**F)F-F5F(F.F(F/F(F(**F)F-F5F(F2F(F3F(F0**F)F- F6F(F.F(F/F(F(**F)F-F6F(F2F(F3F(F0*(I#a0GF$F(F)F-F3F(F(*(F=F(F)F-F/F(F 0**F)F(F5F(F6F(F3F(\"\"%**F)F(F5F(F6F(F/F(!\"%*(F=F(F)F(F3F(F0*(F=F(F) F(F/F(F(*(F'F(I$rhoGF$F(F.F(F(*(F'F(FFF(F2F(F0*(FFF(F5F(F.F(F0*(FFF(F5 F(F2F(F(*(FFF(F6F(F.F(F0*(FFF(F6F(F2F(F(*(F5F(F6F(F3F(F**(F5F(F6F(F/F( F-F(,&F.F(F2F0F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Entry21TermLamb daUn5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Entry21TermLambdaUn 4G6\",$*&,&*&I%t111GF$\"\"\"I)alpha121GF$F*F**&I%t121GF$F*I)alpha111GF $F*F*F*I\"hGF$F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Entry21TermL ambdaUn3G6\",$*&,&*&I%t110GF$\"\"\"I)alpha121GF$F*F**&I%t120GF$F*I)alp ha111GF$F*F*F*I\"hGF$F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Entry 21TermLambdaUn2G6\"**,Z**I\"hGF$\"\"\"I\"qGF$\"\"#I%t111GF$F)I)alpha12 1GF$F)F)**F(F)F*F+I%t121GF$F)I)alpha111GF$F)!\"\"**F*F+I%t010GF$F)F,F) F-F)F1**F*F+F3F)F/F)F0F)F)**F*F+I%t020GF$F)F,F)F-F)F1**F*F+F6F)F/F)F0F )F)**F*F+I%t110GF$F)F/F)F0F)F)**F*F+F9F)F/F)F-F)F1**F*F+F,F)I%t120GF$F )F0F)F)**F*F+F,F)FI5Entry21TermLambdaUn1G6\"**,dr**I\"hGF$\"\"\"I\"qGF$\" \"$I%t111GF$F)I)alpha121GF$F)!\"\"**F(F)F*F+I%t121GF$F)I)alpha111GF$F) F)**F*F+I%t010GF$F)F,F)F-F)F)**F*F+F3F)F0F)F1F)F.**F*F+I%t020GF$F)F,F) F-F)F)**F*F+F6F)F0F)F1F)F.*(I#a1GF$F)F*F+F1F)F)*(F9F)F*F+F-F)F.*(I#a2G F$F)F*F+F1F)\"\"#*(FF&,&*&,,F$F&F5F1F6F1FF&F&F.FN *&F3F&,,F$F&F5F1F6F1FF&F0F&F1*&,,F$F&F5F1F6 F1FF&F&F&I\"qGF%F@F&*&,,*&,&*&,(*&,&F0FNF3F@F&F >F&F&F.F&*&F3F&,*F$F&F5FNF6FNF/F1F&F&F&F7F&F&*&,&*&,*F$F&F5FNF6FNFF&F1F&F?F@F&*&,&*&,(*&,&F0\"\"%F3!\"%F&F>F&F&F.F[p*&F3F &,,F$F&F5FNF6FNFF&,&*&,,F$F&F5FNF6FNFF&F&F.FE*&F3F&,,F$F&F5F NF6FNFF&,(FXFN*&,,F$F&F5FNF6FNFF&F1*&F0F&FaqF&F&F&F>F&F1F&F?F@F&*&,(F^qF@*&,&F.F&*&F3F&,(F5F&F6F &F<#F1F@F&FNF&F7F&F&*&,(FdqF1*&,(F5F&F6F&F/F\\rF&F0F&F&F=F\\rF&F>F&F@F &F?F&F&*&,&F3F&F_qF1F&F7F@F&*&,(FRF&F.F1*&F3F&,(F5F&F6F&FF&F&*&,(F5F&F6F&F/F1F&F0F&F&F=F1F&F>F&F1F&,&Fen F&F1F&FN,&F7F&F>F1F1,&F?F&F1F&F[pF1" }}{PARA 11 "" 1 "" {XPPMATH 20 ", 6*(I%t010G6\"\"\"\"I%t020GF%F&I'lambdaGF%!\"#!\"\"*&I%P112GF%F&F(F*F&* (I%t111GF%F&I%t121GF%F&,&F(F&F*F&!\"%F**&,&*&I%t110GF%F&F/F&F&*&F.F&I% t120GF%F&F&F&F0!\"$F**&,(*&F$F&F'F&F**&I)tinfty10GF%F&I)tinfty20GF%F&F &F,F*F&F0F)F**&F,F&F0F*F**&,&F,F*I#a0GF%F*F&F(F*F&*&,*F;F*F " 0 "" {MPLTEXT 1 0 53 "rho:=factor(solv e(Entry21TermLambdaMinusqCube,rho));\n" }{MPLTEXT 1 0 23 "simplify(rho -(-p*mu));\n" }{MPLTEXT 1 0 39 "simplify(Entry21TermLambdaMinusqCube); \n" }{MPLTEXT 1 0 63 "Lqter:=2*mu*(p-P1(q)/2+h/2/q+2*h/2/(q-1))-h*nu0 \+ -h*nuMinus1*q;\n" }{MPLTEXT 1 0 30 "factor(simplify(Lqbis-Lqter));" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$rhoG6\",$*,I\"p GF$\"\"\"I\"qGF$F(,&F)F(!\"\"F(\"\"#,&I)alpha111GF$F(I)alpha121GF$F+F( ,&I%t111GF$F(I%t121GF$F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I&LqterG6\",(*,,&I)alpha111GF$\"\"\"I)alpha121GF$!\"\"F)I\"qGF$F),&F, F)F+F)\"\"#,&I%t111GF$F)I%t121GF$F+F+,.I\"pGF$F)*&,&I%t010GF$F)I%t020G F$F)F)F,F+#F+F.*&,&F0F)F1F)F)F-!\"#F8*&,&I%t110GF$F)I%t120GF$F)F)F-F+F 8*&I\"hGF$F)F,F+#F)F.*&FAF)F-F+F)F)F.**FAF)F'F)F-F.F/F+F+*,FAF)F'F)F-F )F/F+F,F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Lp:=simplify(-Entry21TermLambdaMinusq/h):\n " }{MPLTEXT 1 0 55 "Eq5:=simplify(Entry21TermLambdaMinusqSquare-(-p*h* Lq));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$Eq5G6\",$ *.I\"hGF$\"\"\",&I)alpha111GF$F(I)alpha121GF$!\"\"F(,0*&I\"pGF$\"\"#I \"qGF$\"\"'#F(\"\"$*&,**$F/F0#!\"%F4*&,,F'F(I%t110GF$#F,F4I%t120GF$F=I %t010GF$F=I%t020GF$F=F(F/F(F(I#a0GF$F3I#a1GF$F3F(F1\"\"&F(*&,.F7F0*&,0 F'#!#5F4I%t111GF$F=F>F(I%t121GF$F=F?#\"\"%F4F@FLFF,FKFTF?!\"#F@FUF< F,F(F/F(F(*&FJF(F>F(F3*&FF3FKF=F?FLF@FLFF=FKF3F(FJF(F(FWF=FAF 8FBF=FOF3F(F1F0F(*&,(*&,(F'F3F?F=F@F=F(F/F(F(FNF8FAF3F(F1F(F(FNF3F(,&F 1F(F,F(FU,&FJF(FKF,F,F1F,F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 649 "a0 := factor(-(p^2*q^6+h*p*q^5-4*p^2*q^5-p*q^5*t010-p*q^5*t020-p* q^5*t110-p*q^5*t120-4*h*p*q^4+h*q^4*tinfty10+6*p^2*q^4+4*p*q^4*t010+4* p*q^4*t020+3*p*q^4*t110-p*q^4*t111+3*p*q^4*t120-p*q^4*t121+q^4*tinfty1 0*tinfty20+6*h*p*q^3-2*h*q^3*tinfty10-4*p^2*q^3-6*p*q^3*t010-6*p*q^3*t 020-3*p*q^3*t110+2*p*q^3*t111-3*p*q^3*t120+2*p*q^3*t121-2*q^3*t010*t02 0+q^3*t110*t121+q^3*t111*t120-2*q^3*tinfty10*tinfty20-4*h*p*q^2+h*q^2* tinfty10+p^2*q^2+4*p*q^2*t010+4*p*q^2*t020+p*q^2*t110-p*q^2*t111+p*q^2 *t120-p*q^2*t121+5*q^2*t010*t020-q^2*t110*t121-q^2*t111*t120+q^2*t111* t121+q^2*tinfty10*tinfty20+h*p*q-p*q*t010-p*q*t020-4*q*t010*t020+t010* t020)/(q*(q^2-2*q+1)));\n" }{MPLTEXT 1 0 2 "a1" }{MPLTEXT 1 0 639 " := factor( (p^2*q^6+h*p*q^5-4*p^2*q^5-p*q^5*t010-p*q^5*t020-p*q^5*t110-p* q^5*t120-4*h*p*q^4+h*q^4*tinfty10+6*p^2*q^4+4*p*q^4*t010+4*p*q^4*t020+ 3*p*q^4*t110-p*q^4*t111+3*p*q^4*t120-p*q^4*t121+q^4*tinfty10*tinfty20+ 5*h*p*q^3-2*h*q^3*tinfty10-4*p^2*q^3-6*p*q^3*t010-6*p*q^3*t020-3*p*q^3 *t110+2*p*q^3*t111-3*p*q^3*t120+2*p*q^3*t121-2*q^3*t010*t020+q^3*t110* t121+q^3*t111*t120-2*q^3*tinfty10*tinfty20-2*h*p*q^2+h*q^2*tinfty10+p^ 2*q^2+4*p*q^2*t010+4*p*q^2*t020+p*q^2*t110-p*q^2*t111+p*q^2*t120-p*q^2 *t121+5*q^2*t010*t020-q^2*t110*t121-q^2*t111*t120+q^2*t111*t121+q^2*ti nfty10*tinfty20-p*q*t010-p*q*t020-4*q*t010*t020+t010*t020)/(q*(q^2-2*q +1)))\n" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 2 "a2" }{MPLTEXT 1 0 673 ":= factor(-(p^2*q^6+h*p*q^5-4*p^2*q^5-p*q^5*t010-p*q^5*t020-p*q^5*t11 0-p*q^5*t120-3*h*p*q^4+h*q^4*tinfty10+6*p^2*q^4+4*p*q^4*t010+4*p*q^4*t 020+3*p*q^4*t110-p*q^4*t111+3*p*q^4*t120-p*q^4*t121+q^4*tinfty10*tinft y20+3*h*p*q^3-3*h*q^3*tinfty10-4*p^2*q^3-6*p*q^3*t010-6*p*q^3*t020-3*p *q^3*t110+2*p*q^3*t111-3*p*q^3*t120+2*p*q^3*t121-q^3*t010*t020+q^3*t11 0*t121+q^3*t111*t120-3*q^3*tinfty10*tinfty20-h*p*q^2+3*h*q^2*tinfty10+ p^2*q^2+4*p*q^2*t010+4*p*q^2*t020+p*q^2*t110-p*q^2*t111+p*q^2*t120-p*q ^2*t121+3*q^2*t010*t020-q^2*t110*t121-q^2*t111*t120+q^2*t111*t121+3*q^ 2*tinfty10*tinfty20-h*q*tinfty10-p*q*t010-p*q*t020-3*q*t010*t020-q*tin fty10*tinfty20+t010*t020)/(q*(q^2-2*q+1)));\n" }{MPLTEXT 1 0 15 "simpl ify(Eq1);\n" }{MPLTEXT 1 0 15 "simplify(Eq2);\n" }{MPLTEXT 1 0 14 "sim plify(Eq5);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#a0G6\",$*(,^q*&I\"pGF$ \"\"#I\"qGF$\"\"'\"\"\"*(I\"hGF$F-F)F-F+\"\"&F-*&F)F*F+F0!\"%*(F)F-F+F 0I%t010GF$F-!\"\"*(F)F-F+F0I%t020GF$F-F5*(F)F-F+F0I%t110GF$F-F5*(F)F-F +F0I%t120GF$F-F5*(F/F-F)F-F+\"\"%F2*(F/F-F+F=I)tinfty10GF$F-F-*&F)F*F+ F=F,*(F)F-F+F=F4F-F=*(F)F-F+F=F7F-F=*(F)F-F+F=F9F-\"\"$*(F)F-F+F=I%t11 1GF$F-F5*(F)F-F+F=F;F-FD*(F)F-F+F=I%t121GF$F-F5*(F+F=F?F-I)tinfty20GF$ F-F-*(F/F-F)F-F+FDF,*(F/F-F+FDF?F-!\"#*&F)F*F+FDF2*(F)F-F+FDF4F-!\"'*( F)F-F+FDF7F-FQ*(F)F-F+FDF9F-!\"$*(F)F-F+FDFFF-F**(F)F-F+FDF;F-FT*(F)F- F+FDFIF-F**(F+FDF4F-F7F-FN*(F+FDF9F-FIF-F-*(F+FDFFF-F;F-F-*(F+FDF?F-FK F-FN*(F/F-F)F-F+F*F2*(F/F-F+F*F?F-F-*&F)F*F+F*F-*(F)F-F+F*F4F-F=*(F)F- F+F*F7F-F=*(F)F-F+F*F9F-F-*(F)F-F+F*FFF-F5*(F)F-F+F*F;F-F-*(F)F-F+F*FI F-F5*(F+F*F4F-F7F-F0*(F+F*F9F-FIF-F5*(F+F*FFF-F;F-F5*(F+F*FFF-FIF-F-*( F+F*F?F-FKF-F-*(F/F-F)F-F+F-F-*(F)F-F+F-F4F-F5*(F)F-F+F-F7F-F5*(F+F-F4 F-F7F-F2*&F4F-F7F-F-F-F+F5,&F+F-F5F-FNF5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#a1G6\"*(,\\q*&I\"pGF$\"\"#I\"qGF$\"\"'\"\"\"*(I\"hGF$F,F(F,F*\" \"&F,*&F(F)F*F/!\"%*(F(F,F*F/I%t010GF$F,!\"\"*(F(F,F*F/I%t020GF$F,F4*( F(F,F*F/I%t110GF$F,F4*(F(F,F*F/I%t120GF$F,F4*(F.F,F(F,F*\"\"%F1*(F.F,F *FF,I)tinfty20GF$F,F,*(F.F,F(F,F*FCF/*(F.F,F*FCF>F,!\"#*&F( F)F*FCF1*(F(F,F*FCF3F,!\"'*(F(F,F*FCF6F,FP*(F(F,F*FCF8F,!\"$*(F(F,F*FC FEF,F)*(F(F,F*FCF:F,FS*(F(F,F*FCFHF,F)*(F*FCF3F,F6F,FM*(F*FCF8F,FHF,F, *(F*FCFEF,F:F,F,*(F*FCF>F,FJF,FM*(F.F,F(F,F*F)FM*(F.F,F*F)F>F,F,*&F(F) F*F)F,*(F(F,F*F)F3F,F<*(F(F,F*F)F6F,F<*(F(F,F*F)F8F,F,*(F(F,F*F)FEF,F4 *(F(F,F*F)F:F,F,*(F(F,F*F)FHF,F4*(F*F)F3F,F6F,F/*(F*F)F8F,FHF,F4*(F*F) FEF,F:F,F4*(F*F)FEF,FHF,F,*(F*F)F>F,FJF,F,*(F(F,F*F,F3F,F4*(F(F,F*F,F6 F,F4*(F*F,F3F,F6F,F1*&F3F,F6F,F,F,F*F4,&F*F,F4F,FM" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#a2G6\",$*(,`q*&I\"pGF$\"\"#I\"qGF$\"\"'\"\"\"*(I\"hGF $F-F)F-F+\"\"&F-*&F)F*F+F0!\"%*(F)F-F+F0I%t010GF$F-!\"\"*(F)F-F+F0I%t0 20GF$F-F5*(F)F-F+F0I%t110GF$F-F5*(F)F-F+F0I%t120GF$F-F5*(F/F-F)F-F+\" \"%!\"$*(F/F-F+F=I)tinfty10GF$F-F-*&F)F*F+F=F,*(F)F-F+F=F4F-F=*(F)F-F+ F=F7F-F=*(F)F-F+F=F9F-\"\"$*(F)F-F+F=I%t111GF$F-F5*(F)F-F+F=F;F-FE*(F) F-F+F=I%t121GF$F-F5*(F+F=F@F-I)tinfty20GF$F-F-*(F/F-F)F-F+FEFE*(F/F-F+ FEF@F-F>*&F)F*F+FEF2*(F)F-F+FEF4F-!\"'*(F)F-F+FEF7F-FQ*(F)F-F+FEF9F-F> *(F)F-F+FEFGF-F**(F)F-F+FEF;F-F>*(F)F-F+FEFJF-F**(F+FEF4F-F7F-F5*(F+FE F9F-FJF-F-*(F+FEFGF-F;F-F-*(F+FEF@F-FLF-F>*(F/F-F)F-F+F*F5*(F/F-F+F*F@ F-FE*&F)F*F+F*F-*(F)F-F+F*F4F-F=*(F)F-F+F*F7F-F=*(F)F-F+F*F9F-F-*(F)F- F+F*FGF-F5*(F)F-F+F*F;F-F-*(F)F-F+F*FJF-F5*(F+F*F4F-F7F-FE*(F+F*F9F-FJ F-F5*(F+F*FGF-F;F-F5*(F+F*FGF-FJF-F-*(F+F*F@F-FLF-FE*(F/F-F+F-F@F-F5*( F)F-F+F-F4F-F5*(F)F-F+F-F7F-F5*(F+F-F4F-F7F-F>*(F+F-F@F-FLF-F5*&F4F-F7 F-F-F-F+F5,&F+F-F5F-!\"#F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "V:=Matrix(3,3,0): \n" }{MPLTEXT 1 0 11 "V[1,1]:=1:\n" }{MPLTEXT 1 0 11 "V[1,2]:=1:\n" } {MPLTEXT 1 0 10 "V[1,3]:=0:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "V[ 2,1]:=0:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "V[2,2]:=1:\n" } {MPLTEXT 1 0 11 "V[2,3]:=1:\n" }{MPLTEXT 1 0 13 "V[3,1]:=1/q:\n" } {MPLTEXT 1 0 17 "V[3,2]:=1/(q-1):\n" }{MPLTEXT 1 0 18 "V[3,3]:=1/(q-1) ^2:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 3 "V;\n" }{MPLTEXT 1 0 18 "B:= Matrix(3,1,0):\n" }{MPLTEXT 1 0 13 "B[1,1]:=h*p:\n" }{MPLTEXT 1 0 48 " B[2,1]:=h*p*q +t010*t020-tinfty10*(tinfty20+h):\n" }{MPLTEXT 1 0 47 "B [3,1]:=p^2-(P1(q)-h*1/q-h*2/(q-1))*p+tdP2(q):\n" }{MPLTEXT 1 0 2 "B;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 38 "VectorC:=simplify(Multiply(V^(- 1),B)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6(-I$ mtdGF$6(-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'norm alF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGF:/%+c olumnspanGF:F4-F56(-F86%Q\"0F'F;F>FAFDFFFHFJFAFDFF-F26(FLF4F4FAFDFF-F2 6(-F56(-I&mfracGF$6)F7-F,6#-I#miGF$6&Q\"qF'/%'italicGQ%trueF'F;/F?Q'it alicF'/%.linethicknessGF:/%+denomalignGQ'centerF'/%)numalignGFco/%)bev elledGQ&falseF'F;FAFDFFFHFJ-F56(-FX6)F7-F,6#-F,6%Ffn-I#moGF$6.Q(&minus ;F'F;F>/%&fenceGFho/%*separatorGFho/%)stretchyGFho/%*symmetricGFho/%(l argeopGFho/%.movablelimitsGFho/%'accentGFho/%'lspaceGQ,0.2222222emF'/% 'rspaceGFeqF7F_oFaoFdoFfoF;FAFDFFFHFJ-F56(-FX6)F7-F,6#-I%msupGF$6%-F#6 %F_pF;F>-F86%Q\"2F'F;F>/%1superscriptshiftGFPF_oFaoFdoFfoF;FAFDFFFHFJF AFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEFco/FGQ'|frleft|hrF'/%/alignme ntscopeGF\\o/%,columnwidthGQ%autoF'/%&widthGFds/%+rowspacingGQ&1.0exF' /%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF_t/%&fra meGF_t/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFho/%-equalcolumnsGF ho/%-displaystyleGFho/%%sideGQ&rightF'/%0minlabelspacingGF\\tF;F>/%%op enGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$ 67-I$mtrGF$6&-I$mtdGF$6(-F,6%-I#miGF$6&Q\"hF'/%'italicGQ%trueF'/%+fore groundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6-Q1&InvisibleTimes ;F'/FDQ'normalF'/%&fenceGQ&falseF'/%*separatorGFN/%)stretchyGFN/%*symm etricGFN/%(largeopGFN/%.movablelimitsGFN/%'accentGFN/%'lspaceGQ&0.0emF '/%'rspaceGFgn-F:6&Q\"pF'F=F@FC/%)rowalignGQ!F'/%,columnalignGF_o/%+gr oupalignGF_o/%(rowspanGQ\"1F'/%+columnspanGFfoF]oF`oFbo-F26&-F56(-F,6' -F,6'F9FFFjnFF-F:6&Q\"qF'F=F@FC-FG6.Q\"+F'F@FJFLFOFQFSFUFWFY/FfnQ,0.22 22222emF'/FinFhp-F,6%-F:6&Q%t010F'F=F@FCFF-F:6&Q%t020F'F=F@FC-FG6.Q(&m inus;F'F@FJFLFOFQFSFUFWFYFgpFip-F,6%-F#6%-F,6%F9Fdp-F:6&Q)tinfty20F'F= F@FCF@FJFF-F:6&Q)tinfty10F'F=F@FCF]oF`oFboFdoFgoF]oF`oFbo-F26&-F56(-F, 6+-F,6#-I%msupGF$6%Fjn-I#mnGF$6%Q\"2F'F@FJ/%1superscriptshiftGQ\"0F'Fb q-F,6%-F#6%-F,6+-I&mfracGF$6)-F,6#-F,6%F\\qFdpF_q-F,6#Fap/%.linethickn essGFfo/%+denomalignGQ'centerF'/%)numalignGFft/%)bevelledGFNF@Fdp-Fjs6 )-F,6#-F,6%-F:6&Q%t111F'F=F@FCFdp-F:6&Q%t121F'F=F@FC-F,6#-Fjr6%-F#6%-F ,6%FapFbq-F]s6%FfoF@FJF@FJF\\sF`sFbtFdtFgtFitF@Fdp-Fjs6)-F,6#-F,6%-F:6 &Q%t110F'F=F@FCFdp-F:6&Q%t120F'F=F@FC-F,6#F]vFbtFdtFgtFitF@Fbq-Fjs6)-F ,6#F9F`tFbtFdtFgtFitF@Fbq-Fjs6)-F,6%F\\sFFF9F]wFbtFdtFgtFitF@F@FJFFFjn Fdp-Fjs6)Fjp-F,6#-Fjr6%FapF\\sF`sFbtFdtFgtFitF@Fdp-Fjs6)-F,6%FauFFFdu- F,6#-Fjr6%F[v-F]s6%Q\"4F'F@FJF`sFbtFdtFgtFitF@Fdp-Fjs6)-F,6#-F,6%-F,6% FgvFFFduFdp-F,6%FauFFFjv-F,6#-Fjr6%F[v-F]s6%Q\"3F'F@FJF`sFbtFdtFgtFitF @F]oF`oFboFdoFgoF]oF`oFbo/%&alignGQ%axisF'/F^oQ)baselineF'/FaoFft/FcoQ '|frleft|hrF'/%/alignmentscopeGF?/%,columnwidthGQ%autoF'/%&widthGFez/% +rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%, columnlinesGF`[l/%&frameGF`[l/%-framespacingGQ,0.4em~0.5exF'/%*equalro wsGFN/%-equalcolumnsGFN/%-displaystyleGFN/%%sideGQ&rightF'/%0minlabels pacingGF][lF@FJ/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "factor(simplify(VectorC[1,1]-C01));\n" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 29 "simplify(VectorC[2,1]-C11));\n" } {MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 28 "simplify(VectorC[3,1]-C12)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Lp:=simplify(Lp);\n" }{MPLTEXT 1 0 100 "Lpbis :=mu*(p*diff(P1(q),q)+h*p*(1/q^2+2/(q-1)^2)-diff(tdP2(q),q)-C01/q^2-C1 1/(q-1)^2-2*C12/(q-1)^3)\n" }{MPLTEXT 1 0 28 "+h*nuMinus1*p-h*c11/(q-1 )^2:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "factor(simplify(Lp-Lpbis) );" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#LpG6\"**,2*(I\"pGF$\"\"#,&I)alp ha111GF$\"\"\"I)alpha121GF$!\"\"F,I\"qGF$\"\"(!\"$**,.I\"hGF$F,F(#!#8F )I%t010GF$F.I%t020GF$F.I%t110GF$F.I%t120GF$F.F,F*F,F(F,F/\"\"'!\"#*(F* F,,(*$F(F)\"#A*&,0F4!\"(F7\"\")F8FDF9F0I%t111GF$F.F:F0I%t121GF$F.F,F(F ,F,*&,&F4F,I)tinfty20GF$F,F,I)tinfty10GF$F,F,F,F/\"\"&F.*(F*F,,(F?F;*& ,0F4F1F7\"\"%F8FPF9\"\"$FEF.F:FQFFF.F,F(F,F,FGF,F,F/FPFQ*&,**&,&F+FCF- F0F,F(F)F,*(,0F4F,F7#!\")FKF8FXF9F.FE#FQFKF:F.FFFZF,F*F,F(F,FK*&,,*&,& FJF1FFF.F,F4F,F,*&,&F:F,FFF,F,FEF,F,*&FJF,FIF,F1*&F7F,F8F,F,*&F9F,FFF, F,F,F+F,F,*&F-F,,,*&,&FJF,FE#F,FQF,F4F,F,*&,&F:#F.FQFFFeoF,FEF,F,F[oF, F\\oFeoF]oFeoF,FQF,F/FQF,*&,**&F(F)F*F,F,*(F*F,,0F4F,F7F " 0 "" {MPLTEXT 1 0 14 "Hamiltonian := " }{MPLTEXT 1 0 82 "mu*(p^2-P1(q)*p+h*p*(1/q+2/(q-1)) +tdP2(q) )-h* nu0*p-h*nuMinus1*q*p -h*c11/(q-1) \n" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 42 "nuMinus1*(t010*t020-tinfty10*(tinfty20+h))" }{MPLTEXT 1 0 1 " " }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "factor(seri es(" }{MPLTEXT 1 0 43 "simplify(Lp-(-diff(Hamiltonian,q))),h=0));\n" } {MPLTEXT 1 0 35 "simplify(Lq-(diff(Hamiltonian,p)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 234 65 "Computation of td\{L\} and verif ication of the gauge transformation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(checkL[1,1]);\n" }{MPLTEXT 1 0 47 "checkL11: =-Q2(lambda)/(lambda-0)/(lambda-1)^2;\n" }{MPLTEXT 1 0 32 "simplify(ch eckL[1,1]-checkL11);\n" }{MPLTEXT 1 0 23 "simplify(checkL[1,2]);\n" } {MPLTEXT 1 0 46 "checkL12:=(lambda-q)/(lambda-0)/(lambda-1)^2;\n" } {MPLTEXT 1 0 31 "simplify(checkL[1,2]-checkL12);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "checkL22bis:=P1(lambda)+Q2(lambda)" }{MPLTEXT 1 0 25 "/(lambda-0)/(lambda-1)^2;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 35 " simplify(checkL[2,2]-checkL22bis);\n" }{MPLTEXT 1 0 53 "checkL21bis:=h *diff( Q2(lambda)/(lambda-q),lambda) +\n" }{MPLTEXT 1 0 123 "L[2,1]*(l ambda-0)*(lambda-1)^2/(lambda-q)-P1(lambda)*Q2(lambda)/(lambda-q)-Q2(l ambda)^2/(lambda-q)/(lambda-0)/(lambda-1)^2:" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 34 "simplify(checkL[2,1]-checkL21bis);" }}{PARA 11 "" 1 " " {XPPMATH 20 "*,I\"pG6\"\"\"\"I\"qGF$F%,&F&F%!\"\"F%\"\"#I'lambdaGF$F (,&F*F%F(F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)checkL11G6\"*,I\"p GF$\"\"\"I\"qGF$F',&F(F'!\"\"F'\"\"#I'lambdaGF$F*,&F,F'F*F'!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "* (I'lambdaG6\"!\"\",&F#\"\"\"F%F'!\"#,&F#F'I\"qGF$F%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)checkL12G6\"*(I'lambdaGF$!\"\",&F&\"\"\"F'F)!\"#,& F&F)I\"qGF$F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,checkL22bisG6\",**&,&I%t010GF$\"\"\"I%t020GF$F)F) I'lambdaGF$!\"\"F)*&,&I%t111GF$F)I%t121GF$F)F),&F+F)F,F)!\"#F)*&,&I%t1 10GF$F)I%t120GF$F)F)F1F,F)*,I\"pGF$F)I\"qGF$F),&F9F)F,F)\"\"#F+F,F1F2F ," }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G1:=Matrix(2,2 ,0):\n" }{MPLTEXT 1 0 12 "G1[1,1]:=1:\n" }{MPLTEXT 1 0 12 "G1[1,2]:=0: \n" }{MPLTEXT 1 0 9 "G1[2,1]:=" }{MPLTEXT 1 0 8 "tinfty10" }{MPLTEXT 1 0 14 "*lambda+eta0:\n" }{MPLTEXT 1 0 12 "G1[2,2]:=1:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "dG1dlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 89 "for i from 1 to 2 do for j from 1 to 2 do dG1dlambda[i,j]:=diff( G1[i,j],lambda): od: od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 88 "Ltilde:=simplify(Multiply(Multiply(G1,checkL),G1^(-1) )+h*Multiply(dG1dlambda,G1^(-1))):\n" }{MPLTEXT 1 0 39 "series(Ltilde[ 1,1],lambda=infinity,2);\n" }{MPLTEXT 1 0 39 "series(Ltilde[1,2],lambd a=infinity,2);\n" }{MPLTEXT 1 0 39 "series(Ltilde[2,2],lambda=infinity ,2);\n" }{MPLTEXT 1 0 39 "series(Ltilde[2,1],lambda=infinity,2);\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 49 "tinfty10+t010+t020+t110+t120-Coh erenceEquation1;\n" }{MPLTEXT 1 0 82 "factor(-residue(Ltilde[2,1]/lamb da,lambda=infinity)+CoherenceEquation1*tinfty10);\n" }{MPLTEXT 1 0 4 " eta0" }{MPLTEXT 1 0 68 ":=factor(solve(factor(-residue(Ltilde[2,1],lam bda=infinity)),eta0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "eta0" } {MPLTEXT 1 0 30 "theo:=1/(tinfty10-tinfty20)*(\n" }{MPLTEXT 1 0 107 "- (2*0*P022+1*P132) +(0^2*C01+1^2*C11+2*1*C12) -h*p*q^2 -tinfty10*((t111 +t121)+0*(t010+t020)+1*(t110+t120))\n" }{MPLTEXT 1 0 45 "+tinfty10*(ti nfty10-tinfty20-h)*(q-1*0-2*1)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "factor(simplify(" }{MPLTEXT 1 0 4 "eta0" }{MPLTEXT 1 0 1 "-" } {MPLTEXT 1 0 4 "eta0" }{MPLTEXT 1 0 59 "theo+1/(t010+t020+t110+t120+2* tinfty10)*CoherenceEquation1*" }{MPLTEXT 1 0 4 "eta0" }{MPLTEXT 1 0 4 "theo" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 84 "simplify(-residue(Lti lde[2,1]/lambda,lambda=infinity)+tinfty10*CoherenceEquation1);\n" } {MPLTEXT 1 0 48 "simplify(-residue(Ltilde[2,1],lambda=infinity));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"OG6$%*protectedGI(_syslibG6\"6#*$I'lambdaGF'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"OG6$%*protectedGI(_syslibG6\"6#*$I'lambdaGF'! \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"OG6$%*protectedGI(_syslibG6\" 6#*$I'lambdaGF'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I\"OG6$%*protec tedGI(_syslibG6\"6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I)tinfty2 0G6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%eta0G6\"**,fq*&I\"pGF$\"\"#I\"qGF$\"\"'\"\"\"*&F(F)F* \"\"&!\"%*(F(F,F*F.I%t010GF$F,!\"\"*(F(F,F*F.I%t020GF$F,F2*(F(F,F*F.I% t110GF$F,F2*(F(F,F*F.I%t120GF$F,F2*&F(F)F*\"\"%F+*(F(F,F*F:F1F,F:*(F(F ,F*F:F4F,F:*(F(F,F*F:F6F,\"\"$*(F(F,F*F:I%t111GF$F,F2*(F(F,F*F:F8F,F>* (F(F,F*F:I%t121GF$F,F2*&F*F:I)tinfty10GF$F)F,*&F(F)F*F>F/*(F(F,F*F>F1F ,!\"'*(F(F,F*F>F4F,FH*(F(F,F*F>F6F,!\"$*(F(F,F*F>F@F,F)*(F(F,F*F>F8F,F K*(F(F,F*F>FCF,F)*(F*F>F6F,FEF,F2*(F*F>F@F,FEF,F2*(F*F>F8F,FEF,F2*(F*F >FCF,FEF,F2*&F*F>FEF)F/*&F(F)F*F)F,*(F(F,F*F)F1F,F:*(F(F,F*F)F4F,F:*(F (F,F*F)F6F,F,*(F(F,F*F)F@F,F2*(F(F,F*F)F8F,F,*(F(F,F*F)FCF,F2*(F*F)F1F ,F4F,F,*(F*F)F6F,FCF,F,*(F*F)F6F,FEF,F)*(F*F)F@F,F8F,F,*(F*F)F@F,FCF,F ,*(F*F)F@F,FEF,F)*(F*F)F8F,FEF,F)*(F*F)FCF,FEF,F)*&F*F)FEF)F.*(F(F,F*F ,F1F,F2*(F(F,F*F,F4F,F2*(F*F,F1F,F4F,!\"#*(F*F,F6F,FCF,F2*(F*F,F6F,FEF ,F2*(F*F,F@F,F8F,F2*(F*F,F@F,FEF,F2*(F*F,F8F,FEF,F2*(F*F,FCF,FEF,F2*&F *F,FEF)Fao*&F1F,F4F,F,F,F*F2,&F*F,F2F,Fao,,F1F,F4F,F6F,F8F,FEF)F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 234 112 "Expression of the Lax matrices in the geometric gau ge after the symplectic reduction and the Painlev\351 2 equation" } {TEXT 234 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 212 183 "Simplification \+ of the formulas after the reduction and expression of the Lax matrices in the geometric gauge after reduction. In this case, we have \\check \{q\}=q and \\check\{p\}=p=\\td\{p\}." }{TEXT 212 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "t111:=1/2*t:\n" }{MPLTEXT 1 0 15 "a lpha111:=1/2:\n" }{MPLTEXT 1 0 16 "alpha121:=-1/2:\n" }{MPLTEXT 1 0 13 "t121:=-t111:\n" }{MPLTEXT 1 0 13 "t020:=-t010:\n" }{MPLTEXT 1 0 13 "t120:=-t110:\n" }{MPLTEXT 1 0 21 "tinfty20:=-tinfty10:\n" } {MPLTEXT 1 0 11 "q:=checkq:\n" }{MPLTEXT 1 0 11 "p:=checkp:\n" } {MPLTEXT 1 0 30 "simplify(CoherenceEquation1);\n" }{MPLTEXT 1 0 20 "si mplify(Trace(L));\n" }{MPLTEXT 1 0 19 "simplify(Trace(A));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 25 "simplify(Trace(checkL));\n" }{MPLTEXT 1 0 24 "simplify(Trace(checkA));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*,I\"hG6\"\"\"\",(*&I'checkqGF%F&I' lambdaGF%F&\"\"$*$F*\"\"#!\"#F)!\"\"F&F*F/,&F*F&F/F&F/,&F*F/F)F&F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "**,&I'checkqG6\"\"\"\"!\"\"F&F&,(*&I'c heckpGF%F&F$\"\"#F+*&F*F&F$F&!\"#*&I\"hGF%F&I'lambdaGF%F&F&F&I\"tGF%F' ,&F0F'F$F&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 ",$*(I\"hG6\"\"\"\",&I'checkqGF%F&!\"\"F&F&I\"tGF%F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Ltilde12:=Ltilde[1,2];\n" }{MPLTEXT 1 0 42 "factor(simplify(Ltilde[2,2]+Ltilde[1,1]));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "eta0bis:=-1/2/tinfty20*( checkq* (" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "q-1)^2*" }{MPLTEXT 1 0 5 " check" }{MPLTEXT 1 0 12 "p^2 -t010^2/" }{MPLTEXT 1 0 5 "check" } {MPLTEXT 1 0 9 "q-t^2/4/(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 23 "q -1)^2-t/4*(4*t110+t)/(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 6 "q-1)+ (" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 16 "q-2)*tinfty20^2\n" } {MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 21 "factor(eta0-eta0bis);" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Ltilde12G6\"*(I'lambdaGF$!\"\",&F&\" \"\"F'F)!\"#,&F&F)I'checkqGF$F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(eta0bisG6\",$*&I)tinfty10GF$! \"\",,*(I'checkqGF$\"\"\",&F+F,F(F,\"\"#I'checkpGF$F.F,*&I%t010GF$F.F+ F(F(*&I\"tGF$F.F-!\"##F(\"\"%*(F3F,,&I%t110GF$F6F3F,F,F-F(F5*&,&F+F,F4 F,F,F'F.F,F,#F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Ltilde111Order2:=factor(simplify(re sidue(Ltilde[1,1]*(lambda-1),lambda=1))):\n" }{MPLTEXT 1 0 66 "Ltilde1 11Order1:=factor(simplify(residue(Ltilde[1,1],lambda=1))):\n" } {MPLTEXT 1 0 66 "Ltilde110Order1:=factor(simplify(residue(Ltilde[1,1], lambda=0))):\n" }{MPLTEXT 1 0 13 "Ltilde[1,1]:\n" }{MPLTEXT 1 0 94 "Lt ilde11bis:= Ltilde111Order2/(lambda-1)^2+Ltilde111Order1/(lambda-1)+Lt ilde110Order1/lambda:\n" }{MPLTEXT 1 0 35 "simplify(Ltilde[1,1]-Ltilde 11bis);\n" }{MPLTEXT 1 0 21 "Ltilde111Order2bis:=(" }{MPLTEXT 1 0 5 "c heck" }{MPLTEXT 1 0 11 "q-1)*eta0+(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 6 "q-1)*(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p*" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 4 "q^2-" }{MPLTEXT 1 0 5 "check" } {MPLTEXT 1 0 2 "p*" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 13 "q+tinfty 10):\n" }{MPLTEXT 1 0 46 "simplify(Ltilde111Order2-Ltilde111Order2bis) ;\n" }{MPLTEXT 1 0 21 "Ltilde111Order1bis:=-" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 8 "q*eta0 -" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p* " }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 3 "q*(" }{MPLTEXT 1 0 5 "check " }{MPLTEXT 1 0 17 "q-1)^2-tinfty10:\n" }{MPLTEXT 1 0 46 "simplify(Lti lde111Order1-Ltilde111Order1bis);\n" }{MPLTEXT 1 0 20 "Ltilde110Order1 bis:=" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "q*eta0+" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p*" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 3 "q*(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 8 "q-1)^2:\n" } {MPLTEXT 1 0 45 "simplify(Ltilde110Order1-Ltilde110Order1bis);" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Ltilde211Order2:=factor(simplify(residue(Ltilde[2,1]* (lambda-1),lambda=1))):\n" }{MPLTEXT 1 0 66 "Ltilde211Order1:=factor(s implify(residue(Ltilde[2,1],lambda=1))):\n" }{MPLTEXT 1 0 66 "Ltilde21 0Order1:=factor(simplify(residue(Ltilde[2,1],lambda=0))):\n" }{MPLTEXT 1 0 94 "Ltilde21bis:= Ltilde211Order2/(lambda-1)^2+Ltilde211Order1/(l ambda-1)+Ltilde210Order1/lambda:\n" }{MPLTEXT 1 0 35 "simplify(Ltilde[ 2,1]-Ltilde21bis);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 21 "Ltilde211 Order2bis:=(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 12 "q-1)*((eta0+" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p*" }{MPLTEXT 1 0 5 "check" } {MPLTEXT 1 0 3 "q*(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 27 "q-1)+ti nfty10)^2-(t111^2)/(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 9 "q-1)^2) :\n" }{MPLTEXT 1 0 46 "simplify(Ltilde211Order2-Ltilde211Order2bis);\n " }{MPLTEXT 1 0 21 "Ltilde211Order1bis:=-" }{MPLTEXT 1 0 5 "check" } {MPLTEXT 1 0 10 "q*( (eta0+" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 3 " p*(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 18 "q-1)^2)^2 -t010^2/" } {MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "q^2) :\n" }{MPLTEXT 1 0 46 "si mplify(Ltilde211Order1-Ltilde211Order1bis);\n" }{MPLTEXT 1 0 20 "Ltild e210Order1bis:=" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 10 "q*( (eta0+" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 3 "p*(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 18 "q-1)^2)^2 -t010^2/" }{MPLTEXT 1 0 5 "check" } {MPLTEXT 1 0 6 "q^2):\n" }{MPLTEXT 1 0 45 "simplify(Ltilde210Order1-Lt ilde210Order1bis);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dqdt:=1/h*Lq:\n" }{MPLTEXT 1 0 24 " dpdt:=simplify(1/h*Lp):\n" }{MPLTEXT 1 0 17 "dcheckqdt:=dqdt;\n" } {MPLTEXT 1 0 1 "d" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 10 "pdt:=dpdt ;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "dG1dt:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 91 "for i from 1 to 2 do for j from 1 to 2 do dG1dt[i,j]: =diff(G1[i,j],t)+diff(G1[i,j],checkq)*" }{MPLTEXT 1 0 9 "dcheckqdt" } {MPLTEXT 1 0 14 "+diff(G1[i,j]," }{MPLTEXT 1 0 6 "checkp" }{MPLTEXT 1 0 2 ")*" }{MPLTEXT 1 0 9 "dcheckpdt" }{MPLTEXT 1 0 11 ": od: od:\n" } {MPLTEXT 1 0 82 "Atilde:=simplify(Multiply(Multiply(G1,checkA),G1^(-1) )+h*Multiply(dG1dt,G1^(-1))):" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*dcheckqdtG6\"*(I\"hGF$!\"\",(*(I'checkpGF$\"\"\"I'chec kqGF$F+,&F,F+F'F+\"\"#F.*&F&F+F,F.F+*&F&F+F,F+F'F+I\"tGF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*dcheckpdtG6\",$*,,2*&I'checkpGF$\"\"#I'chec kqGF$\"\"(!#7*&,&*$F)F*\"#_*&F)\"\"\"I\"hGF$F3!\")F3F+\"\"'F3*&,(F0!#) )*&F4F3F)F3\"#G*&,&F4F3I)tinfty10GF$!\"\"F3F>F3!\"%F3F+\"\"&F3*&,(F0\" #sF:!#OF<\"#7F3F+\"\"%F3*&,0F0!#GF2\"#?*&F4F3F>F3F-*$I\"tGF$F*F?*&FNF3 I%t110GF$F3F@*$I%t010GF$F*F@*$F>F*FFF3F+\"\"$F3*&,0F0FGF2F@FLFGFMF?FOF GFQFFFSF@F3F+F*F3*&F+F3FRF*F-FQFGF3F4F?,&F+F3F?F3!\"$FNF?F+!\"##F3FG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Atilde12:=simplify(Atilde [1,2]);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 35 "simplify(Atilde[1,1]+A tilde[2,2]);\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I)Atilde12G6\"*(,&I'checkqGF$\"\"\"!\"\"F(F(,&I'lambdaGF$F(F)F(F)I\"tG F$F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(I\"hG6\"\"\"\",&I'checkqGF%F &!\"\"F&F&I\"tGF%F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "At ilde111Order2:=factor(simplify(residue(Atilde[1,1]*(lambda-1),lambda=1 )));\n" }{MPLTEXT 1 0 1 "A" }{MPLTEXT 1 0 65 "tilde111Order1:=factor(s implify(residue(Atilde[1,1],lambda=1))):\n" }{MPLTEXT 1 0 1 "A" } {MPLTEXT 1 0 65 "tilde110Order1:=factor(simplify(residue(Atilde[1,1],l ambda=0)));\n" }{MPLTEXT 1 0 23 "Atilde11OrderConstant:=" }{MPLTEXT 1 0 63 "factor(-simplify(residue(Atilde[1,1]/lambda,lambda=infinity)));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "A" }{MPLTEXT 1 0 12 "tilde[1,1] :\n" }{MPLTEXT 1 0 41 "Atilde11bis:= Atilde111Order1/(lambda-1)+" } {MPLTEXT 1 0 21 "Atilde11OrderConstant" }{MPLTEXT 1 0 2 ":\n" } {MPLTEXT 1 0 35 "simplify(Atilde[1,1]-Atilde11bis);\n" }{MPLTEXT 1 0 21 "Atilde11OrderConstant" }{MPLTEXT 1 0 16 "bis:=-tinfty10*(" } {MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 8 "q-1)/t:\n" }{MPLTEXT 1 0 9 "si mplify(" }{MPLTEXT 1 0 21 "Atilde11OrderConstant" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 21 "Atilde11OrderConstant" }{MPLTEXT 1 0 3 "bis" } {MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 21 "Atilde111Order1bis:=-" } {MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 6 "q-1)*(" } {MPLTEXT 1 0 5 "eta0+" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p*" } {MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 4 "q^2-" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p*" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 13 "q+tinft y10)/t" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 45 "simplify(Atilde111Orde r1-Atilde111Order1bis);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Atilde221Order2:=factor(simplify (residue(Atilde[2,2]*(lambda-1),lambda=1)));\n" }{MPLTEXT 1 0 66 "Atil de221Order1:=factor(simplify(residue(Atilde[2,2],lambda=1))):\n" } {MPLTEXT 1 0 66 "Atilde220Order1:=factor(simplify(residue(Atilde[2,2], lambda=0))):\n" }{MPLTEXT 1 0 87 "Atilde22OrderConstant:=factor(-simpl ify(residue(Atilde[2,2]/lambda,lambda=infinity)));\n" }{MPLTEXT 1 0 64 "Atilde22bis:= Atilde221Order1/(lambda-1)+Atilde22OrderConstant:\n" }{MPLTEXT 1 0 35 "simplify(Atilde[2,2]-Atilde22bis);\n" }{MPLTEXT 1 0 40 "Atilde22OrderConstantbis:=(tinfty10-h)*(" }{MPLTEXT 1 0 5 "check " }{MPLTEXT 1 0 8 "q-1)/t:\n" }{MPLTEXT 1 0 58 "simplify(Atilde22Order Constant-Atilde22OrderConstantbis);\n" }{MPLTEXT 1 0 21 "Atilde221Orde r1bis:=(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 11 "q-1)*(eta0+" } {MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p*" }{MPLTEXT 1 0 5 "check" } {MPLTEXT 1 0 4 "q^2-" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p*" } {MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 15 "q+tinfty10)/t:\n" }{MPLTEXT 1 0 45 "simplify(Atilde221Order1-Atilde221Order1bis);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0Atilde111Order2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0Atilde110Order1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6Atilde11OrderConstantG6\",$*(,&I'checkqGF$\"\"\"!\"\"F)F)I)tinf ty10GF$F)I\"tGF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0Atil de221Order2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6Atilde22Order ConstantG6\",$*(,&I'checkqGF$\"\"\"!\"\"F)F),&I\"hGF$F)I)tinfty10GF$F* F)I\"tGF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Atilde211Order2:=factor(simp lify(residue(Atilde[2,1]*(lambda-1),lambda=1)));\n" }{MPLTEXT 1 0 66 " Atilde211Order1:=factor(simplify(residue(Atilde[2,1],lambda=1))):\n" } {MPLTEXT 1 0 66 "Atilde210Order1:=factor(simplify(residue(Atilde[2,1], lambda=0)));\n" }{MPLTEXT 1 0 87 "Atilde21OrderConstant:=factor(-simpl ify(residue(Atilde[2,1]/lambda,lambda=infinity)));\n" }{MPLTEXT 1 0 13 "Atilde[2,1]:\n" }{MPLTEXT 1 0 42 "Atilde21bis:= Atilde211Order1/(l ambda-1):\n" }{MPLTEXT 1 0 35 "simplify(Atilde[2,1]-Atilde21bis);\n" } {MPLTEXT 1 0 22 "Atilde211Order1bis:=-(" }{MPLTEXT 1 0 5 "check" } {MPLTEXT 1 0 14 "q-1)/t*((eta0+" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 2 "p*" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 3 "q*(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 25 "q-1)+tinfty10)^2-t^2/4)-(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 5 "q-2)*" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 8 "q*t/(4*(" }{MPLTEXT 1 0 5 "check" }{MPLTEXT 1 0 7 "q-1)):\n" } {MPLTEXT 1 0 14 "factor(series(" }{MPLTEXT 1 0 51 "simplify(Atilde211O rder1-Atilde211Order1bis),p=0));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 " " {XPPMATH 20 ">I0Atilde211Order2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0Atilde210Order1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6Atilde21OrderConstantG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }